On the Theory of n-Polynomial ϕ-Convex Functions

This paper introduces a novel extension of n-polynomial convex functions, referred to as “n-polynomial ϕ-convex functions”. The proposed concept generalizes the existing notion of n-polynomial convexity introduced in the recent literature. Within this new structural framework, we derive several novel inequalities and show that the absolute derivative of a function can characterise this specific convexity class. Our results show that employing the Hölder– (Formula presented.) can and enhanced power-mean integral inequalities yields significantly sharper and more efficient bounds compared to those obtained via standard Hölder and power-mean approaches. Furthermore, we provide explicit examples, algorithmic simulations, and visual graphs to validate the theoretical attributes of the definition. The proposed framework is expected to contribute to the development of integral inequality theory and special functions.